Problem

Source: 2012 Indonesia Round 2.5 TST 2 Problem 4

Tags: floor function, number theory unsolved, number theory



The sequence $a_i$ is defined as $a_1 = 1$ and \[a_n = a_{\left\lfloor \dfrac{n}{2} \right\rfloor} + a_{\left\lfloor \dfrac{n}{3} \right\rfloor} + a_{\left\lfloor \dfrac{n}{4} \right\rfloor} + \cdots + a_{\left\lfloor \dfrac{n}{n} \right\rfloor} + 1\] for every positive integer $n > 1$. Prove that there are infinitely many values of $n$ such that $a_n \equiv n \mod 2012$.